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Theorem bj-eltag 33289
Description: Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-eltag (𝐴 ∈ tag 𝐵 ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-eltag
StepHypRef Expression
1 df-bj-tag 33287 . . 3 tag 𝐵 = (sngl 𝐵 ∪ {∅})
21eleq2i 2831 . 2 (𝐴 ∈ tag 𝐵𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3 elun 3896 . 2 (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}))
4 bj-elsngl 33280 . . 3 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
5 0ex 4942 . . . 4 ∅ ∈ V
65elsn2 4356 . . 3 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
74, 6orbi12i 544 . 2 ((𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}) ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
82, 3, 73bitri 286 1 (𝐴 ∈ tag 𝐵 ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1632  wcel 2139  wrex 3051  cun 3713  c0 4058  {csn 4321  sngl bj-csngl 33277  tag bj-ctag 33286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324  df-bj-sngl 33278  df-bj-tag 33287
This theorem is referenced by: (None)
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